were no resistance the velocity a grain would attain after any period of time of subsidence would depend only on its specific gravity and would increase as the specific gravity increases. The size of the fragment is immaterial. When the opposition of the fluid is considered it is seen that a large grain will fall faster than a small one of the same kind and also that grains of high specific gravity will fall faster than ones of less specific gravity. What are called free settling ratios are those of relative sizes at which particles of different material will have the same settling rate. Free settling ratios are conventionally expressed in terms of quartz. The free settling ratio of galena is 4.01. This would be the same as saying that a grain of galena of 4.01 mm. diameter would settle at the same rate as a quartz grain of 16.08 diameter etc. If screens of these two sizes were employed to effect a grading or any pair having the ratio as multiplier or divisor all the grains of galena could be separated from all the grains of quartz by jigging.
Where there is a mass of ore in a fluid such as water the combined ore and water must be considered as a "fluid" and the specific gravity of this "fluid" must be used in making computations. This favors the settlement and separation of heavy grains smaller than those indicated by the free settling ratios. It will be evident further that since the heavier grains are hindered by their neighbors in taking a direct path to the screen, their average velocity in reaching the screen is very much less than their free settling velocities and hence the relative settlement rates will be more directly in proportion to specific gravity. Owing to this and other factors Richards by experimental work has determined that the true settling ratios are higher than the free settling ratios. He has determined for example that the ratio for galena to quartz is 5.842. He denominates these ratios of increased amount "hindered settling ratios." The accompanying curve shows the hindered settling ratios for common minerals. If two heavy materials are to be separated from one another as well as from quarts then the hindered settling ratio of the lighter of the two heavy minerals must be divided into that of the heavier to find the screen ratio, if the hindered settling ratio of the quartz and the lighter of the two heavy minerals is less than this ratio. For example the working screen ratio for a mixture of chalcocite, galena and quartz is not 5.842 or 3.115 but 1.875, the ratio of the hindered settling factors of the two heavy minerals.
The mode of operation of a fixed sieve jig with pure mineral fragments and screened within the limits imposed by hindered settling ratios may be gained from the following elementary description of the separation action. If two grains, Fig. 2, G and M of different volumes and different specific gravities, M being of greater specific gravity and either the same size or smaller within the limits imposed by hindered settling, be at equal distance above the screen of the jig as at points A and A and they be given an upward pulsion by a fluid such as water, then they ought to arrive at the end of the pulsion at some such position as BB, but since both fall to some extent and M falls faster than G at the end of the upstroke, G will be in some position such as C, and M as some position as D. On the downward return following the upward movement, both grains will be aided by their downward impelling force, but since this gives to M the greater velocity, it will take some such position as E and G, F. By a number of cycles as just described M will ultimately reach the screen, while G will still be at some distance above it. If now while falling relatively to one another, the two grains are subjected to a sidewise current, such as is produced by the flowage of ore and water through a jig, then evidently G will be more affected laterally than M, and may be caused to flow entirely away and to waste, while M though not reaching the screen in a position vertically under the point where it first enters the jigging compartment where the screen is, but ahead of it, can be drawn through the holes in the screen or